Representations of Data

Small bits of data can yield interesting patterns, such as the
spirals in nature that give rise to the golden mean.
We will use our creative inquiry
techniques
to analyze larger data sets with many dimensions of data
for patterns. Work in the same group that you created your
stock market data and graph.
Class Data
Click on this Class Data Excel File.
The computer should then download the file.
Open the file classdata.xls from Excel.
You will see an Excel table filled in - I took the data
from the sheets you filled in, and put it into Excel.
Average or Mean
To calculate the average of a bunch of numbers, we
add them all up, and then divide by how many numbers we have.

Mental Rotations Test (MRT) I graded these on a scale of 0, 1, or 2.
A 2 means that you scored both choices correctly (choices a and d),
while 1 means that one of your
choices was correct, while the other was incorrect.

Click on Q1, which is empty. Type in
=average(n2:n16)
and hit Enter. This will calculate the average of the female
scores on the Timed MRT. What is this average?

Click on Q2. Type in
=average(n17:n28)
and hit Enter. This will calculate the average of the male scores on the
Timed MRT. What is this average?

Click on Q3, which is empty. Calculate the average of the female
scores on the Untimed MRT. What is this average?

Click on Q4. Calculate the average of the male scores on the
Untimed MRT. What is this average?

Height and Armspan

Click on Q7. Calculate the class average of armspan using
=average(h2:h28)
What is this average?

Click on Q8. Calculate the class average of height.
What is this average?

Mean, Median and Mode as Measures of CenterMean/AverageTo calculate the average of a bunch of numbers, we
add them all up, and then divide by how many numbers we have.
MedianTo calculate the median,
we first put our data in increasing order, and then find the middle number
and place. If there is no such middle number, we take the 2 numbers
next to the middle place, and take the average of them.
For example,
0,0,1,3 has an average of 1, since we would take (0+0+1+3)/4 = 1.
It has a median of .5, since the middle place is between the 2nd 0 and the 1.
Since there is no actual number in this place, we take the average of 0 and 1,
which is .5.
The median is both a place and a number.
ModeTo calculate the mode, we take the most frequently occurring
value. For example, 0 would be the mode of 0,0,1,3 since it occurs
more frequently than any other value.
Mean, median and mode are all different measures of the data's
"center", and
certain circumstances make each better to use at times.

Number of Siblings

Click on Q10. Calculate the class average of the
number of siblings. What is this average?
Here I have reordered the sibling
data.
0
0
0
1
1
1
1
1
1
1
1
1
1
[the median place and number lies here between two data points]
1
[the average lies here between two data points]
2
2
2
2
2
2
2
3
3
3
3
4
Notice that
the mode of the number of siblings is 1
because it occurs the most times.
Notice that the average is higher than the median
since the data is not weighted evenly about the middle place.
If you think of the median place as the middle of a scale, then the lower
numbers (0s and 1s) pull the average down
while the higher numbers pull it up. While there are an equal number
of items on
either side of this scale, the average equals
the median only when these pulls cancel each other out, if the
data is evenly weighted about the median.
This does not happen in this case.
The 0s and 1s pull the average down
less than the higher numbers pull it up, since the median matches many of
the numbers in the lower numbers (so they have no pull at all).
What is a reasonable measure of center in this case? Well, the median
is the actual center of the data and so it
is a good measure of center. In this case it tells us that half the class
has that many or less siblings.
Notice that while the average is not a number in our data set,
and it is not the center of the data, it does tell us about the
distribution of the data.
Finally, the mode just tells us the most frequently occurring
number in the data set.

Distance from Home

Click on Q11. Type in
=median(d2:d27)
What is this median?

Click on Q12. Type in
=mode(d2:d27)
What is this mode?

Calculate the class average of the distance
from home in Q13. What is this average?

Take out your paper copy of the class data and turn to the side
where it is ordered by distance from home.
Mark where the
average and median are,
(just like I did for the median and average with the
number of siblings).

Explain why the average is higher than the median by using the idea of
a scale balancing about the median.

Make sure that we have checked over your numerical answers.
Then, under File, release on Close, and click on Don't Save. Open up your
stock market data that you sent yourself last week (do not delete your email
- we will use it again!).
Stock Market Data

The Mean and Median of VolumeVolume is the number of shares
bought or sold in a given day.

What Excel box contains the first data entry for Volume?

What Excel box contains the last data entry for Volume?

Click on g2, which is empty, and calculate the
average of Volume. What is this value?

Calculate the median of Volume
in G3. What is this value?

On the stock update packet
turn to the page with the graph on it. The volume is represented as
a bar graph on the bottom. To read the volume on a given day, you go to the
top of the corresponding
rectangle, go over to the left of the graph, and then read the
number from there.
Draw a horizontal line
at the point on the y-axis which matches the
median of Volume on this graph. This may or may not actually
correspond to a specific day that attained this volume.
Show me so that I
can mark off that you have done this correctly.

The Mean and Median of HighHigh is the highest price that
a stock hits each day.

What is the average of the data in the High column?

What is the median of the data in the High column?

What is the smallest High value? You can use a command like
=QUARTILE(D2:D66,0) where you change 66 to the number of rows of data you have.
The quartiles are the 25% markers, so the 0th quartile is the lowest
data point, the 1st quartile = q1 is the 25% marker, etc.

What is the largest High value?
You can use a command like =QUARTILE(D2:D66,4)

Analyses and Connections

Many studies have shown that
there are gender differences on the Timed MRT in that men perform better,
but that women perform just as well on the untimed MRT.
It makes sense that people (men or women)
would perform better under less time pressure
(although it just depends on the person).
Does our class data follow these patterns?

Search the web for
Lawrence Summers. Then find at least two different perspectives about the
performance of women in mathematics (for example, one from Summers and the
other from someone else). Summarize the perspectives in your own words.

Search the web and/or texts
to find information about Leonardo da Vinci's speculation
about the relationship between armspan and height. Summarize what you found
in your own words.

Search the web to find information about the ape index and its
relationship to rock climbing and summarize what you found.
Then identify yourself on the class data list (but don't tell
anyone else which row you are in!) and calculate your own index:

Recall that the median is the middle place after we put the data in
increasing
order and that if we think of this as the center of a scale, then
we can see whether the mean is higher or lower by seeing whether the data
tips the scale to one side or another (you may wish to review my
explanation in the Sibling section above).
Even if the data is not in increasing order (on your stock graph for Volume
it is not),
we can still use the idea of the median as the center of a scale to see
whether the data tips the scale higher or lower than the median.
Use only the Volume bars on your stock graph and your knowledge of
mean compared to median to discuss why you can see from these bars whether
the mean is above or below the median.

Many of you may have very similar
numbers for the mean and median of High,
even though your stock fluctuated up and down from one
day to the next. Explain what is going on using your knowledge of the
difference between mean and median, the relatively short time span
of the data, generalities about the price ranges of stocks in this kind of
time span,
and your knowledge of the stock market from past hw readings.